t6_to_t7_route_equivalence
plain-language theorem explainer
From φ-forcing (T6), the direct canonical-period path to the eight-tick law (T7) and the indirect path T6→T8→T7 via dimensional forcing yield the same T7 surface. Anyone assembling the unified T0–T8 chain cites this to treat the two routes as one construction. The proof is a structure inhabitant: it plugs in the existing T6→T7 and T6→T8→T7 bridges and discharges agreement fields by reflexivity.
Claim. Assume $\varphi$ is forced: $\varphi^2=\varphi+1$, $\varphi>0$, and $\varphi$ is the unique positive solution of that equation. Then the direct canonical-period bridge from T6 to the eight-tick statement (T7) and the composite bridge T6 $\to$ dimensional forcing (T8, $D=3$) $\to$ eight-tick (T7) produce identical T7 data: both witness $8=2^3$ and the from-dimension packaging by the same canonical equalities.
background
The module UnifiedForcingChain aims at a complete inevitability chain: every landmark from the absolute floor through T0–T8 is forced from the Recognition Composition Law plus normalization and calibration, rather than merely shown compatible.
T6 asserts that in a discrete ledger with self-similar cost, the only scaling ratio is $\varphi=(1+\sqrt{5})/2$, characterized by $\varphi^2=\varphi+1$, positivity, and uniqueness among positive reals. T7 is the eight-tick octave (period $2^3=8$). T8 forces spatial dimension $D=3$. An RS-compatible dimension packages linking, eight-tick synchronization ($2^D=8$), gap-sync, and cellular/acyclicity conditions; dimension_unique already shows any such $D$ equals 3 from linking alone.
Two routes connect T6 to T7: a direct canonical-period bridge, and an indirect path that first forces $D=3$ then recovers the eight-tick law. The route-equivalence certificate records both and asserts they land on the same T7 surface.
proof idea
Term-mode structure construction for T6_To_T7_RouteEquivalence h6.
direct_routeis filled by the existing canonical bridget6_to_t7_canonical_bridge_holds h6.via_t8_dim_bridgeis filled byt6_to_t8_dimension_bridge_holds h6.via_t8_to_t7_bridgeappliest8_to_t7_bridge_holdsto the T8 component of that dimension bridge.- The three agreement fields (
t7_eight_is_2_cubed_agrees,t7_from_dimension_agrees,canonical_period_agrees_at_D3) are purerfl: both routes compute the same canonical equalities at $D=3$ and $8=2^3$.
No new forcing arithmetic; the lemma only packages and equates already-proved bridges.
why it matters
In the forcing chain, T6 pins $\varphi$, T7 is the eight-tick octave, and T8 is $D=3$. Historically those steps can look like alternate stories (period from self-similarity versus period from $2^D$ after dimension forcing). This certificate collapses them: the canonical-period route and the dimension-bridge route are the same universal construction up to definitional agreement.
It feeds complete_forcing_chain, the top-level unconditional package that threads T-1 through T8 via successive bridges. Without route equivalence, the chain would carry two competing T7 witnesses; with it, the assembler may pick either path and still quote a single T7 surface. Framework landmarks touched: T6 ($\varphi$ forced), T7 (eight-tick, period $2^3$), T8 ($D=3$), and the RS-compatible dimension package (linking, eight-tick, gap-sync).
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